Cut the tortillas into 6 wedges. Pour enough oil into a large skillet to cover by 1/8 of an inch. Heat over medium-high heat. When the oil in hot, drop the cut tortillas into the pan, working in batches so as not to over-crowd the pan. Fry until golden brown on one side, 1-2 minutes. Flip and fry on the other side until golden brown, another 1-2 minutes. Transfer to a paper towel lined plate to drain and sprinkle with a little salt. Repeat with remaining tortillas until they are all fried. Drain off all but 1 tablespoon of oil.
Add the red enchilada sauce and cook for 2 minutes. Reduce the heat to medium-low. Whisk the eggs with ½ teaspoon salt and ½ teaspoon pepper. Pour into the hot sauce and cook, stirring occasionally, until the eggs are fully incorporated into the sauce and set, about 10 minutes. Add the tortillas back into the pan and stir to make sure they are fully coated with sauce.
Top with the cilantro and cheese and serve immediately.